Discussion and Conclusions
Eq. (1) can be solved effectively by the homotopy perturbation method [24-28] and the variational approach [2, 29]. By the semi-inverse method [30, 31], we can establish a variational formulation for Eq. (1), which is
\(J\left(u\right)=\int_{0}^{T/4}\left\{\frac{1}{2}{\dot{x}}^{2}-\frac{1}{2}x^{2}+\frac{1}{2}\ln\left(1+x^{2}\right)\right\}\text{dt}\)(25)
By comparison also with those in open literatures, we conclude that He’s frequency is the simplest while its accuracy is also extremely high. Considering the simplest calculation, He’s frequency formulation greatly promote the development of the nonlinear science, especially the nonlinear vibration.